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Stable generalized complex structures. (English) Zbl 1395.58020

A generalized complex structure is a triple \((M, \mathbb{J},H)\), \(M\) a smooth \(n\)-manifold, \(\mathbb{J}\) is a complex structure on \(\mathbb{T}M=TM\oplus T^ast M\), orthogonal for the split signature matrix on this bundle, whose \(+i\)-eigenbundel \(L\) is involutive for the Courant bracket twisted by \(H\) [M. Gualtieri, Ann. Math. (2) 174, No. 1, 75–123 (2011; Zbl 1235.32020)].
Generalized complex structures \((M,\mathbb{J},H)\) and \((M',\mathbb{J}',H')\) are said to be equivalent if there is a diffeomorphism \(\phi\) and 2-form \(b\) such that \(\phi^\ast H'=H+db\), and \(\mathbb{J}'\circ (\phi^\ast e^b)=(\phi^\ast e^b)\circ \mathbb{J}\), \(e^b:X+\xi\to X+\xi+\iota_X b\). If \(\phi\) is trivial, \(\mathbb{J}\) and \(\mathbb{J}'\) are said to be gauge equivalent. Two generalized complex structure with same undelying pair \((M,H)\) are gauge equivalent when they are conjugate by a B-field gauge symmetry.
On \(M\), a pair \((U,s)\), \(U\) a complex line bundle and \(s\) a section of \(U\) transverse to the 0-section, is referred as a complex divisor (§1. Def.1.1). Following this definition, the canonical line bundle \(K\) of a generalized complex manifold, and related notions are introduced in §2.1. Then the following definition of stable generalized-complex structure is given
Definiton 2.10. A generalized complex structure is stable when its anticanonical section vanishes transversally, so that \(D=(K^\ast,s)\) defines a complex divisor called the anticanonical divisor.
Applying T-duality, a stable generalized complex structure on the total space of \(U\) on the total space of \(U\) removed the zero section, is constructed. Here \((U,\bar{\partial})\) \(\bar{\partial}\) is derived from \(d+H\wedge\), is a generalized holomorphic line bundle over the stable generalized complex manifold \((U,\mathbb{J})\) [G. R. Cavalcanti and M. Gualtieri, in: A celebration of the mathematical legacy of Raoul Bott. Based on the conference, CRM, Montreal, Canada, June 9–13, 2008. Providence, RI: American Mathematical Society (AMS). 341–365 (2010; Zbl 1200.53062)].
Adopting residue maps derived from logarithm and elliptic tangent bundles, explained in §1, to stable generalized complex structure, equivalence between stable generalized complex structures and complex log symplectic structure is proved (Theorem 3.2). Here Theorem is proved assuming \((M,H)\) are fixed in the generalized complex structures. Then the results are extended to the case varying \(H\) (Theorem 3.7), which shows equivalence with co-oriented elliptic symplectic structures, if we consider only gauge equivalence class of stable structures. These results are used to define two periodic maps, one for deformation in which \(H\) is fixed (§3.2. Periodic map defines a bijection between germs of deformations of complex log symplectic structure \(\sigma\) up to equivalence and germs of smooth paths beginning at the origin in \(H^\ast(M\setminus D,\mathbb{C})\) (Theorem 3.14)), one where is not fixed (§3.3). The periodic map defines a bijection between germs of deformation of \(\omega\), an elliptic symplectic form with zero elliptic residue on the compact manifold with elliptic divisor \((M,|D|)\), up to equivalence and germs of smooth paths beginning at \([\omega]\in H^2(M\setminus D,\mathbb{R})\oplus H^1(D,\mathbb{R})\) (Theorem 3.22)).
In the rest of this Section, the following Darbough type theorem is proved:
Theorem 3.27. Any complex log symplectic form is equivalent , near a point on its degeneracy divisor, to the normal form \[ \sigma_0=d\log w\wedge dz+i\omega. \] Then classification of a tubular neighborhood of \(D\) (Theorem 3.28) and a Lagranian brane neighborhood theorem (Theorem 3.31) are presented.

MSC:

58H15 Deformations of general structures on manifolds
53D18 Generalized geometries (à la Hitchin)
32Q99 Complex manifolds
81T99 Quantum field theory; related classical field theories
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